Optimal. Leaf size=151 \[ -\frac {(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}-\frac {1}{192} (65-1194 x) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{512} (3865-8082 x) \sqrt {3 x^2+5 x+2}+\frac {41053 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1024 \sqrt {3}}-\frac {1325}{128} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {812, 814, 843, 621, 206, 724} \begin {gather*} -\frac {(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}-\frac {1}{192} (65-1194 x) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{512} (3865-8082 x) \sqrt {3 x^2+5 x+2}+\frac {41053 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1024 \sqrt {3}}-\frac {1325}{128} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 812
Rule 814
Rule 843
Rubi steps
\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx &=-\frac {(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}-\frac {1}{8} \int \frac {(-332-398 x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx\\ &=-\frac {1}{192} (65-1194 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac {1}{768} \int \frac {(40938+48492 x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx\\ &=-\frac {1}{512} (3865-8082 x) \sqrt {2+5 x+3 x^2}-\frac {1}{192} (65-1194 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}-\frac {\int \frac {-2525724-2955816 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{36864}\\ &=-\frac {1}{512} (3865-8082 x) \sqrt {2+5 x+3 x^2}-\frac {1}{192} (65-1194 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac {41053 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{1024}-\frac {6625}{128} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {1}{512} (3865-8082 x) \sqrt {2+5 x+3 x^2}-\frac {1}{192} (65-1194 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac {41053}{512} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )+\frac {6625}{64} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {1}{512} (3865-8082 x) \sqrt {2+5 x+3 x^2}-\frac {1}{192} (65-1194 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac {41053 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{1024 \sqrt {3}}-\frac {1325}{128} \sqrt {5} \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 120, normalized size = 0.79 \begin {gather*} \frac {159000 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )+205265 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-\frac {2 \sqrt {3 x^2+5 x+2} \left (6912 x^5-28512 x^4-80064 x^3-118996 x^2+40412 x+293973\right )}{2 x+3}}{15360} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.85, size = 121, normalized size = 0.80 \begin {gather*} \frac {41053 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{512 \sqrt {3}}-\frac {1325}{64} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )+\frac {\sqrt {3 x^2+5 x+2} \left (-6912 x^5+28512 x^4+80064 x^3+118996 x^2-40412 x-293973\right )}{7680 (2 x+3)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 149, normalized size = 0.99 \begin {gather*} \frac {205265 \, \sqrt {3} {\left (2 \, x + 3\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 159000 \, \sqrt {5} {\left (2 \, x + 3\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 4 \, {\left (6912 \, x^{5} - 28512 \, x^{4} - 80064 \, x^{3} - 118996 \, x^{2} + 40412 \, x + 293973\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{30720 \, {\left (2 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.17, size = 671, normalized size = 4.44 \begin {gather*} -\frac {41053}{3072} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}{{\left | 2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {1325}{128} \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {325}{128} \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {1304805 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{9} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 2064120 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{8} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 4382950 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{7} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 10490640 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{6} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 19083456 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 33372000 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{4} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 42760170 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 60102000 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{2} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 21448395 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 36498600 \, \sqrt {5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{7680 \, {\left ({\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{2} - 3\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 195, normalized size = 1.29 \begin {gather*} \frac {1325 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{128}+\frac {41053 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\right )}{3072}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{10 \left (x +\frac {3}{2}\right )}-\frac {53 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{20}+\frac {199 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{192}+\frac {1347 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{512}-\frac {265 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{48}-\frac {1325 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{128}+\frac {13 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.46, size = 163, normalized size = 1.08 \begin {gather*} -\frac {1}{20} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {199}{32} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {65}{192} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{4 \, {\left (2 \, x + 3\right )}} + \frac {4041}{256} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {41053}{3072} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {1325}{128} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {3865}{512} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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